We distinguished between fundamental and effective forces between colloids
IUPAC definition:
The attractive or repulsive forces between molecular entities (or between groups within the same molecular entity) other than those due to bond formation or to the electrostatic interaction of ions or of ionic groups with one another or with neutral molecules. The term includes:
- dipole–dipole
- dipole-induced dipole
- London (instantaneous, spontaneous dipole-induced dipole) forces.
The term is sometimes used loosely for the totality of nonspecific attractive or repulsive intermolecular forces.
The electrostatic potential of a charge distribution can be expanded as:
V(\mathbf{r})=\frac{1}{4 \pi \epsilon_0} \sum_{l=0}^{\infty} \sum_{m=-l}^l \frac{4 \pi}{2 l+1} \frac{Q_{l m}}{r^{l+1}} Y_l^m(\theta, \phi)
where Y_l^m(\theta, \phi) are spherical harmonics and Q_{l m}=\int r^{\prime l} Y_l^{m *}\left(\theta^{\prime}, \phi^{\prime}\right) \rho\left(\mathbf{r}^{\prime}\right) d^3 r^{\prime}:
For large distances (r \gg molecular size), the potential is dominated by the lowest non-zero multipole moment.
No need to memorise this.
We are focusing on atoms that are globally neutral. The first non-trivial term is the the dipole.
Static dipolew are asymmetric distribution of charge stationary in time
The interaction energy between two dipoles \mu_1 and \mu_2 separated by a distance r is proportional to:
U(r) \propto-\frac{\mu_1 \mu_2}{r^3}
A permanent dipole can induce a temporary dipole in a nearby neutral atom. The induced dipole then interacts with the permanent dipole.
The induced dipole itself is proportional to the electric field of the incoming dipole, and it is \mu_B \propto \mu_A E_A \propto \dfrac{\mu}{r^3}.
Then the interaction energy is just
U(r) \propto -\frac{\mu^2 \alpha}{r^6}
where \alpha is the polarizability of the neutral atom.
Orbitals provide the average expected electron density at every point in space, \langle\rho(\mathbf{r})\rangle=|\psi(\mathbf{r})|^2
Instantaneous fluctuations from the mean occur in time and depend on density-density correlations. The variance is non-uniform .
This means that even when the density is neutral, on short timescales (femtoseconds, electronic transiiton times) the desnity distributions can be thought of as asymmetric.
These quanto-mechanical fluctuations again produce interactions known as London dispersion forces
London dispersion interaction
\Large U(r)=-\frac{C}{r^6}
The key insight is the 1/r^{6} decay of the interactionwhich decays much mor rapidly than Coulomb’s 1/r.
They are short-range in 3d, since the total energy scales like \sim \int V(r) r^{d-1} d r and 6>3
Take now two spherical colloids of radus R at distance H.
Integrating the London interaction over all volume elements yield the van der Waals attracive potential
Colloid-colloid van der Waals attractive interaction
\Large W_{w d W}(h)=-\frac{A_H}{g} f(h / R)
whera A_H is the Hamaker constant and f(h / R)=\left[\frac{2 R^2}{h^2-4 R^2}+\frac{2 R^2}{h^2}+\ln \left(\frac{h^2-4 R^2}{h^2}\right)\right].
Colloids are often charged. In the solution there will be
A double layer forms due to the accumulation of co-ions near the colloid surface, and repulsion of counter-ions.
Its width depends on the ion concentration in the solvent, which can be adjusted by adding or removing salts.
When two colloids come close, the charges in their double layers will interact giving rise to a repulsive interaction. Not simply Coulombic interaction, as it is mediated by the other (opposite) charges in the medium.
This means in practice taking Poisson’s equation
\nabla^2 \psi(\mathbf{r})=-\frac{\rho_{\mathrm{tot}}(\mathbf{r})}{\varepsilon}= \frac{\rho_{\mathrm{fixed}}(\mathbf{r})+\rho_\mathrm{ions}(\mathbf{r})}{\varepsilon}
and making the ion concentration depend on the electrostatic potential \psi(\mathbf{r}) itself via a Boltzmann weight
\rho_{\mathrm{ions}}(\mathbf{r})= z e n_s \exp \left(-\frac{z e \psi}{k_B T}\right)
with n_s the far-away (bulk) concentration of ions of salt and ze the ion charge.
This yields the nonlinear Poisson-Bolztmann equation
\nabla^2 \psi(\mathbf{r})=-\frac{1}{\varepsilon}\left[\rho_{\text {fixed }}+ ze n_s \exp \left(-\frac{z e \psi}{k_B T}\right)\right] .
No need to memorise this.
Linearisation around room temperature (e\psi\ll k_BT) yields the Debye–Hückel equation
\nabla^2 \psi= \psi/\lambda_D^2
where \lambda_D is yhe Debye length
\lambda_D=\sqrt{\frac{1}{8 \pi \lambda_B n_s}} itself dependent on the Bjerrumm length \lambda_B: the scale at which two elemementary charge have energy k_B T.
The final form of the potential is and exponential repulsion with scale \lambda_D
Important
\Large W_{D R}(h)=B \frac{R}{\lambda_B} \exp \left(-h / \lambda_D\right)
to sum up hat identical colloids in solution have two interactions of opposite sign
a van der Waals components, typically attractive
a double layer component repulsive in nature
The sum of the two gives rise to the DLVO (Derjaguin–Landau–Verwey–Overbeek) interaction which is an elementary model for colloid stability and aggregation.
→ Large energy costs in approaching atoms: steric repulsive interactions below a van der Waals radius.
A phenomenological model for a generic atomistic interaction potential is then
V_{\mathrm{LJ}}(r)=\frac{A_n}{r^n}-\frac{B_m}{r^m}
where the first term is the short range repulsion and the second a generalised attraction.
A convenient choice is the Lennard-Jones potential of scale \sigma (twice the vdW radius)
V_{\mathrm{LJ}}(r)=4 \varepsilon\left[\left(\frac{\sigma}{r}\right)^{12}-\left(\frac{\sigma}{r}\right)^6\right],
where m=6 is chosen to match London’s forces and n=12 is chosen to match experiments and computational convenience.
Atomistic, but at the heart of numerous coarse-grained models .
Steric repulsion (hindrance) means exlcuded volume: at thermal energy scales, an atom cannot be placed at the position of another atom.
Geometrical point of view:
Excluded volume means that, neglecting attractions, atoms/molecules//macromolecules/colloids are spheres that cannot overlap.
Distribute spheres in space:
Exclusion means forbidden configurations.
Statistical mechanics approach: simplest system with the required ingredient, studied in detail.
Macroscopic analogue:
Clothpins on a line
Investigate the probability distribution \rho(x) of finding the centre of a particle at position x. Parameters:
We can treat the problem algorthmically
Algorithm
The key parameter is the packing fraction \phi = \dfrac{dN}{L} (non-dimensional)1.
What do the peaks mean? It means that it is more likely to find the particles close to the walls:
The source of this effect is entropy:
F=U - TS=\cancel{U} - TS
No internal energy, only entropy, so temperature is only a scale parameter.
More realism: Mixture of colloids (yellow) and polymers (squiggly lines inside red circles). The depletion layers are as thick as the polymer radius (red circles) and are indicated with the dashes around the colloids. When the two layers do not overlap, the osmotic pressure due to the polymers on the colloids is balanced. When there is overlap, there is a region inaccessible to the polymers (purple) and the pressure is unbalanced, leading to aggregation.
Depletion forces play a pivotal role in colloidal aggregation.
The basic mechanisms are the same as in our 1D example.
We are going to work thrpugh the details of a particualr model
In the absence of colloids, the entire volume V is accessible to the polymer V_{\rm accessible} = V
Grand potential of the polymers \Omega = -k_BT e^{\mu_p/k_B T}V_{\rm accessible}
Introduce one colloid: the polymers can not get closer than the distance R+\delta
Excluded volume around one polymer
Introduce second colloid at distance r from first colloid.
The accessible volume is V_{\rm accessible}^{\infty}=V-2V_{\rm exclusion}
Two situations:
A second colloid is introduced at a short distance r. The exclusion regions overlap (light blue), increasing the accesible volume.
V_{\mathrm{overlap}}(r)=\dfrac{4 \pi}{3} R_d^3\left[1-\frac{3}{4} \frac{r}{R_d}+\frac{1}{16}\left(\frac{r}{R_d}\right)^3\right]
V_{\rm accessible}^\prime=V-(2V_{\rm exclusion}-V_{\mathrm{overlap}})
\begin{aligned} W_{\rm AO}(r) = \Omega(r)-\Omega^{\infty} & =-k_BT e^{\mu/k_B T}\left(V_{\rm accessible}(r)-V_{\rm accessible}(\infty)\right)\\ & = -k_BT e^{\mu/k_B T}\left[V-2V_{\rm exclusion}+V_{\mathrm{overlap}}(r)-(V-2V_{\rm exclusion})\right]\\ & = -k_BT e^{\mu/k_B T} V_{\mathrm{overlap}(r)}\\ & = -k_B T \rho_p V_{\mathrm{overlap}(r)} \end{aligned}
The result is an attractive depletion potential controlled by the geometry of the overlap
Asakura-Oosawa potential (1954)
\Large W_{\rm AO} (r) = - \dfrac{4 \pi \rho_p^r k_B T}{3} (R+\delta)^3\left[1-\dfrac{3}{4} \dfrac{r}{R+\delta}+\frac{1}{16}\left(\dfrac{r}{R+\delta}\right)^3\right] \quad 2R\leq r< 2R+\delta
| Interaction Type | Description | Range |
|---|---|---|
| Van der Waals | Attractive forces arising from induced dipoles between particles. | Short-range |
| Double Layer | Electrostatic repulsion due to overlapping electrical double layers around charged particles. | Long-range |
| DLVO | Combination of van der Waals attraction and double layer repulsion. | Short and long range |
| Depletion | Typically attractive interactions emerging from purely entropic interactions | Short range |